| L(s) = 1 | − 7-s − 2·11-s + 2·13-s − 19-s − 2·23-s − 5·25-s + 6·29-s + 8·31-s − 6·37-s + 12·43-s − 8·47-s + 49-s − 14·53-s − 8·59-s − 2·61-s + 6·71-s − 2·73-s + 2·77-s + 4·79-s − 16·83-s − 16·89-s − 2·91-s − 2·97-s − 12·101-s + 18·107-s + 6·109-s − 6·113-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 0.603·11-s + 0.554·13-s − 0.229·19-s − 0.417·23-s − 25-s + 1.11·29-s + 1.43·31-s − 0.986·37-s + 1.82·43-s − 1.16·47-s + 1/7·49-s − 1.92·53-s − 1.04·59-s − 0.256·61-s + 0.712·71-s − 0.234·73-s + 0.227·77-s + 0.450·79-s − 1.75·83-s − 1.69·89-s − 0.209·91-s − 0.203·97-s − 1.19·101-s + 1.74·107-s + 0.574·109-s − 0.564·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.072348116622340362415541171425, −7.21305150315764483087390090425, −6.33969081927775734412645760527, −5.91559710311683439820285187631, −4.90788451211076257560752873720, −4.20320976383562817002856830656, −3.26168525541079232864382782586, −2.50490787196561638135161386003, −1.36270374918222285778714784866, 0,
1.36270374918222285778714784866, 2.50490787196561638135161386003, 3.26168525541079232864382782586, 4.20320976383562817002856830656, 4.90788451211076257560752873720, 5.91559710311683439820285187631, 6.33969081927775734412645760527, 7.21305150315764483087390090425, 8.072348116622340362415541171425
